real Real may refer to: Currencies * Argentine real * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Nature and science * Reality, the state of things as they exist, rathe ...

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, Harnack's curve theorem, named after Axel Harnack, gives the possible numbers of connected components that an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

can have, in terms of the degree of the curve. For any algebraic curve of degree in the real

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...

, the number of components is bounded by :

\frac \le c \le \frac+1.\

The maximum number is one more than the maximum

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

of a curve of degree , attained when the curve is nonsingular. Moreover, any number of components in this range of possible values can be attained. Trott bitangents

A curve which attains the maximum number of real components is called an M-curve (from "maximum") – for example, an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

with two components, such as

y^2=x^3-x,

or the

Trott curve In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for wh ...

, a quartic with four components, are examples of M-curves. This theorem formed the background to

Hilbert's sixteenth problem Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topology ...

. In a recent development a

Harnack curve Harnack is the surname of a German family of intellectuals, artists, mathematicians, scientists, theologians and those in other fields. Several family members were executed by the Nazis during the last years of the Third Reich. * Theodosius Harnack ...

is shown to be a curve whose

amoeba An amoeba (; less commonly spelled ameba or amœba; : amoebas (less commonly, amebas) or amoebae (amebae) ), often called an amoeboid, is a type of Cell (biology), cell or unicellular organism with the ability to alter its shape, primarily by ...

has area equal to the

Newton polygon In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the ultrametric field of interest was ''essentially'' the field of f ...

of the

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

, which is called the characteristic curve of

dimer model In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by pl ...

s, and every Harnack curve is the spectral curve of some dimer model.()

References

* Dmitrii Andreevich Gudkov, ''The topology of real projective algebraic varieties'', Uspekhi Mat. Nauk 29 (1974), 3–79 (Russian), English transl., Russian Math. Surveys 29:4 (1974), 1–79 *

Carl Gustav Axel Harnack Carl Gustav Axel Harnack (, Dorpat (now ) – 3 April 1888, Dresden) was a Baltic German mathematician who contributed to potential theory. Harnack's inequality applied to harmonic functions. He also worked on the real algebraic geometry of pla ...

''Ueber die Vieltheiligkeit der ebenen algebraischen Curven''
Math. Ann. 10 (1876), 189–199 * George Wilson, ''Hilbert's sixteenth problem'',

Topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

17 (1978), 53–74 * * {{Citation , last=Mikhalkin , first=Grigory , author-link=Grigory Mikhalkin , title=Amoebas of algebraic varieties, year=2001, mr=2102998, arxiv=math/0108225 , bibcode=2001math......8225M Real algebraic geometry Theorems in algebraic geometry